Data Structures and Algorithms

Lab 6: Labyrinth

Due on Wednesday, October 30th at 11:59 PM. This is a team lab. You and your assigned lab partner(s) will complete this lab together. Make sure that you are familiar with the Partner Etiquette guidelines. You may discuss the concepts of this lab with other classmates, but you may not share your code with anyone other than course staff and your lab partner(s). Do not look at solutions written by students other than your team. If your team needs help, please post on the Piazza forum or contact the instructor or ninjas. If you have any doubts about what is okay and what is not, it’s much safer to ask than to risk violating the Academic Integrity Policy.

Overview

Aerial photograph of a corn maze

Uncle Shuck’s Farm

By UncleShuck’s farm, Dawsonville, GA

This lab comes in three parts:

Implementing stack and queue data structures
Writing an application to solve a maze using (at the user’s option) either breadth-first or depth-first search
Answering written questions regarding algorithmic analysis

The main goals of this lab are to use data structures to implement algorithms and to gain practice using induction. Concepts you will be familiar with after this lab include:

Stacks and queues
Depth-first search
Breadth-first search
Algorithmic analysis

Note: This is a two-week lab assignment. However, the lab is due in three weeks due to the mid-semester break. This lab is not designed to make you work over break, nor are you expected to. There are, however, two weeks' worth of work in this lab, and we do expect you to work on it this week. Plan accordingly.

As usual, you can find a link to your GitHub repository on Teammaker.

Part I: Implementing Stacks and Queues

The first part of this lab involves writing code for a program called maze. This program will solve provided mazes using the depth-first and breadth-first search algorithms discussed in class. To implement these algorithms, you will first need to create implementations of the Stack and Queue ADTs.

Your Starting Code

When you clone your repository, you will see the following files. Files you may need to change appear in bold.

adts/: A directory containing the C++ definitions of the ADTs you will use for this lab (e.g., List). You are not permitted to change these files, but you’ll probably want to refer to them.
linkedQueue.h/linkedQueue-inl.h: The LinkedQueue class, which you will implement.
linkedStack.h/linkedStack-inl.h: The LinkedStack class, which you will implement.
manualTests.cpp: A sandbox test program for you to use while you develop your program.
testStackQueue.cpp and testMaze.cpp: Unit tests for your data structure. These tests have already been written. You will not have to write your own tests for this assignment, but you should definitely run the ones you’ve been given!
position.h/position.cpp: The Position class which keeps track of the contents in a particular position in a maze.
maze.h/maze.cpp: The Maze class which is used to represent a maze and provide solutions to it.
mazeUtils.h/mazeUtils.cpp: Provided utility functions for the Maze class, such as reading files and rendering solutions.
main.cpp: The program which reads a maze, solves it, and prints the result.
test_data/: A directory containing several files for testing your main application. The format is described below.

We can group these files into the major components of the maze lab, which we recommend implementing in the given order:

ADTs and data structures. You are provided with ADTs (List,Stack, Queues, OrderedCollection) in the adts folder and some implementations (STLList). You will implement LinkedQueue and LinkedStack in the corresponding files. You will thoroughly test your implementations using manualTests.cpp and testStackQueue.cpp.
Helper methods for the maze application. These are the provided Position class (which describes locations in a 2D grid), the Maze class (for representing the maze itself) which you will implement, and the provided maze utilities (for loading and drawing your mazes). You will thoroughly test your implementations using manualTests.cpp and testMaze.cpp.
The main application. You will write the main function for the maze program in main.cpp to bring all of the components together.

ADT Implementations

For this and future labs, you will occasionally be provided with implementations of the ADTs we discussed in class. These implementations make use of the C++ standard template library (STL) and are written as classes named e.g. STLList. To use these classes, simply #include the header the same way you normally would, using the directory names as necessary (e.g. #include "adts/stlList.h). You are welcome to inspect these files, but recall that a feature of implementing ADTs is that you only need to know the interface. The STLList, for example, has the exact same interface as your LinkedList implementation from last week:

/* ADTs */
#include "adts/list.h"

/* Implementations */
#include "adts/stlList.h"

int main(){
  List<int>* list = new STLList<int>();
  list->insertFirst(10);
}

Implementing `LinkedQueue` and `LinkedStack`

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Type Hierarchy for Stacks and Queues

The LinkedQueue and LinkedStack classes appear in the like-named files in your starter code. As we discussed in lecture, these classes are easily implemented using a linked list; you can find a linked list implementation in the form of the STLList class in the file adts/stlList.h. You should proceed by implementing each of the methods in the files linkedQueue-inl.h and linkedStack-inl.h. These methods are not meant to be complex; they require very little code to complete. You do not need to throw your own exceptions from these objects' methods since the underlying list will do so for you. For example, while pop() should raise an exception if it is called on an empty list, the STLList will throw this for you when you try to remove an element from an empty list.

This lab includes an OrderedCollection abstract class which describes the commonalities between queues and stacks (also as discussed in lecture). Note that e.g. LinkedQueue is a subclass of both Queue and OrderedCollection, so it will need not only enqueue and dequeue methods but also insert and remove methods. The insert and remove methods can be implemented simply by calling enqueue and dequeue; we will rely on this commonality later to reduce the amount of code you need to write!

Once you’ve completed your LinkedQueue and LinkedStack implementations, run the unit tests (make testStackQueue followed by ./testStackQueue) to see if they are working correctly. After these tests pass, proceed to the Maze part of the lab.

Part II: Solving Mazes

The Layout of a Maze

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An example maze. Black squares are walls.

Once you have completed your LinkedQueue and LinkedStack implementations, you are prepared to implement the search algorithms we discussed in lecture. We begin by considering how we will represent the mazes we intend to solve.

Each maze is a rectangular grid. Each space in the grid can connected to the orthogonally adjacent spaces. That is, from any location in the grid, you can move one space left, right, up, or down; you cannot move diagonally. The starting point of every maze is the upper left corner; the exit is in the bottom right.

The layout of a particular maze will be stored in a text data file with the extension .map; you can find several examples in your test_data directory. We have already written the code which loads files in this format for you; this explanation is just for reference. A map file contains the following:

The width of the maze in squares (that is, the number of columns in the map).
The height of the maze in squares (that is, the number of rows in the map).
For each row of the map, a string describing all of the spaces in that row.
- Each # character is a wall.
- Each . character is an open space.

For instance, the following is a valid map:

5 3
...##
.#...
...#.

And corresponds to the grid pictured.

The `Position` Class

We will represent a maze as a two-dimensional grid of Position objects. Each Position object contains relevant information for one place in the maze: the (X,Y) coordinates (where (0,0) is in the upper-left), whether that position is a wall, and some fields which will be used during the search to construct an appropriate path through the maze. The constructor of the Position class takes the X and Y coordinate values and initialize the other fields to suitable default values (like nullptr or false).

The `Maze` Class

another aerial photo of a different corn maze

Star Wars Maze

By The MAiZE Company

A Maze object represents our knowledge of a particular maze; we can use this information to find a path through the maze using various search strategies. The two-dimensional grid that you create for your maze (in the Maze class) is a member field positions of type Position***: an array of arrays of Position pointers. For example, positions[0][0] will obtain the pointer (Position*) to the Position object in the upper-left corner. You’ll have to use nested for-loops to initialize positions properly.

This class has two public methods — solveBreadthFirst and solveDepthFirst — which will perform the appropriate search algorithm and return an answer. The answer is in the form of a List<Position*>*: a pointer to a new list of positions which constitute a correct path through the maze from start to finish. These two algorithms are very similar: they only differ in which data structure they use to perform the search! Regardless of whether we are using a stack (depth first) or a queue (breadth first), the search algorithm goes as follows:

Add the start position to your data structure
Mark the start position as visited
While there are positions left in the data structure:
- Take a position from the data structure; call it the current position.
- If the current position is the exit, when our search is complete; we can break the loop!
- Otherwise, for each neighbor of the current position:
  - If that neighbor has not been visited:
    
    Mark the neighbor as visited
    
    Record the current position as previous to the neighbor
    
    Add the neighbor to the data structure

At the end of the loop, you can determine whether you found a path by looking at the exit position. If that position has been visited, you just need to follow the previous position pointers backward adding each Position to the List you need to return until you reach the start to describe the path to take. If the exit position has not been visited, then there is no possible path through the maze.

To avoid writing the same code twice, we can implement the above algorithm in the private method solve. This method takes an argument of type OrderedCollection<Position*> and uses that data structure during the search. Then, solveBreadthFirst and solveDepthFirst both call solve, each providing an appropriate data structure. Your solve method will return the list of positions that describe the correct path through the maze (or a nullptr in the event that no such path exists).

Testing the `Maze` Class

Make sure to run the provided unit tests on your code as you develop using make testMaze and ./testMaze; definitely test your code once you’ve finished the Maze class! It’ll be easier to find most bugs in your Maze implementation by direct testing than it will be by trying to find them by hand by running the maze program.

Some of the provided map files in test_data have multiple possible solutions. As a result, your output may differ from the provided solutions depending on how you explore the maze. To get the same results as the provided solutions, you should explore neighboring spaces in the following order in your getNeighbors method: up, left, right, and then down.

Implementing `main`

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A breadth-first path through the maze.

The implementation for main relies mostly on the classes you have already completed and is thus relatively short for such a large lab assignment. The loadMap and renderAnswer functions have been written for you. loadMap will read a map file; it will return a Maze* if load is successful and will throw a runtime_error otherwise. renderAnswer will produce a string containing a solution, which looks like the map from the map file but contains @ characters on the path that you found. For instance, here is a possible execution of the program:

$ ./maze test_data/cycle.map breadth
@@###
#@@@#
#.#@#
#..@@
####@

Your program will take two command-line arguments:

The name of the map file containing the maze to solve
Either "depth" or "breadth" to indicate the type of search to perform

If there is a solution to the provided map, your program should print the rendering of that solution. If there is no solution, your program should print a message to that effect.

Solutions to the provided test mazes are given in the .solution.txt files in the test_data directory:

$ cat test_data/cycle__breadth.solution.txt
@@###
#@@@#
#.#@#
#..@@
####@

If a map is unsolvable, then its solution file will be missing.

Invalid Inputs

Your maze program should gracefully handle the following cases of invalid input:

The user provides an invalid number of command-line arguments.
The user provides a search type other than "depth" or "breadth".
The user provides an invalid maze file (to be handled by catching the exception from loadMap).

Note that you are not permitted to allow your program to crash in these cases; you must catch exceptions and print messages accordingly, using syntax similar to the following:

try {
    ... // do stuff
} catch (runtime_error e) {
    ... // handle error
}

Memory

Your program is required to run without any memory errors. Use valgrind to determine if you have any memory errors; we will do so, and any errors reported by valgrind will result in lost points. You should also use valgrind as a debugging tool as you develop your program.

You are also required not to have any memory leaks, although memory leaks will be scored far more generously. You are encouraged to worry about memory leaks only after your program is working correctly. Once your program works, commit and push it and then consider making changes to solve leaks. It’s much better to have a correct, leaky program than it is to have a leak-free program that crashes.

Coding Style Requirements

As usual, you will also be required to observe good coding practices:

Your C++ code must have proper and consistent indentations.
You must have proper and consistent usage of spacing & braces for if, else, for, and while conditions as well as class definitions and code blocks.

For more information about good style, please see the course style guide.

Part III: Algorithmic Analysis

In this part of the assignment, you will answer questions related to induction and algorithmic analysis. You will submit your answers just as before: in a typeset PDF format. You are encouraged to use LaTeX to typeset your work; we have provided WrittenLab.tex as a starting point. Your solution should appear in WrittenLab.pdf.

For this part of the lab, answer the following questions.

Prove each of the following claims by induction.
- The sum of the first $n$ even numbers is $n^2+n$. That is, $\sum\limits_{i=1}^{n} 2i = n^2 + n$.
- $\sum\limits_{i=1}^{n} \dfrac{2}{3^i} = 1 - \dfrac{1}{3^n}$
- For every $n \geq 1$, $5^n - 1$ is divisible by $4$. In other words, for every $n \geq 1$, there exists some integer constant $z_n$ such that $5^n - 1 = 4z_n$. (Note that $z$ is different for each power of $5$.) You may write your proof either by discussing divisibility by four in general or by discussing specific $z_n$ values.
The function minPos, given below in pseudocode, takes as input an array A of size n of numbers. It returns the smallest positive number in the array. If no positive numbers appear in the array, it returns positive infinity (+∞). (Note that zero is neither positive nor negative). Using induction, prove that the minPos function works correctly. Clearly state your recursive invariant at the beginning of your proof.
```
Function minPos(A,n)
  If n = 0 Then
    Return +∞
  Else
    best <- minPos(A,n-1)
    If A[n-1] < best And A[n-1] > 0 Then
      best <- A[n-1]
    EndIf
    Return best
  EndIf
EndFunction
```

Summary of Requirements

When you are finished, you should have

A working Maze class
A main which provides a command-line interface to solve mazes
The ability to handle bad command-line arguments or user input
No memory errors!
Code which conforms to the style requirements above
Written answers to the algorithmic analysis questions in PDF form
A completed lab assignment survey from each student